无穷远点的留数计算及留数定理的推广

利用求极限的方法可计算复变函数在无穷远点的留数；留数定理可推广到扩充复平面上无界集合的情形和围线所围区域内具有无穷多个奇点或具有非孤立奇点的情形。     

任意阶极点留数的Matlab计算方法

留数又称残数,它是复变函数论中的一个重要概念,但Matlab中的Residue函数仅能处理多项式分式函数.本文首先根据孤立奇点的特点,提出了用Matlab确定极点阶次的方法,然后再根据高阶极点的留数公式,基于符号运算计算该极点处的留数.在此过程中,还解决了Matlab无法直接求解复数点处极限的问题.该方法的优点是不需要事前判断极点阶次,可以计算任意类型函数的留数,使得留数的计算更加简单.     

用极点升级法计算留数

<正> 函数分别是它们的3级极点和m-2级极点.若按照通常的方法来求留数,计算过程是相当繁琐,工作量是很大.请看例:     

谈解析函数级数展开式的一些应用

通过实例给出解析函数的级数展开式在求留数、积分及收敛级数求和中的具体应用.     

留数法在有理函数积分中的应用

留数的思想可在计算有理函数积分时用于确定待定系数，这种确定待定系数的留数法适用于一切有理函数的积分。     

用极点升级法计算留数

<正> 有一类函数,如sinz/z~4芝警,(1-cosz)/z~m(m>2)等等,z=0分别是它们的3级极点和m-2级极点。若按照通常的方法来求留数,这个计算过程是相当繁琐,工作量是很大的。请看例:     

关于极点留数计算规则的一点补充

<正> 一般来说,求函数在其奇点处的留数,只须求出它的罗伦级数中C_(-1)(z-z_0)~(-1)项的系数就可以了。但是对不便写出罗伦级数的函数,在z_0是极点的情形下,下面几个在特殊情况下求留数的规则,是很有用的。     

从历史的角度引入复积分

复积分的概念起源于计算实定积分的问题,根据历史上原有的简单方法解释柯西积分定理和留数定理可以使学生更加深刻地理解其基本内涵.     

一类极点的留数计算

本文探讨了一类比较特殊的极点类型的留数计算并从理论上给出证明.     

商的导数情形下的等价无穷小替换

对于复变函数中高阶极点留数的计算,常会碰到商的导数的极限,给出了商的导数情形下等价无穷小替换的一种方法,简化了计算.     

Topological Hochschild homology of number rings

We calculate an explicit formula for the topological Hochschild homology of a discrete valuation ring of characteristic zero with finite residue field. From this we deduce the topological Hochschild homology of global number rings.

     

An Approach to the Determination of the Resultant of Two Entire Functions

We calculate the sum of the values of an entire function at the zeros of the other entire function by means of the formula of logarithmic residue. As a result, we can answer the question whether these functions have common zeros or not. Thus, we developed an approach to the determination of the resultant of two entire functions.     

Vasiek利率模型下的亚式期权的定价问题和数值分析

本文研究了随机利率满足Vasiek模型时带有浮动的敲定价格的欧式看涨亚式期权的定价问题.通过对所涉及的退化的抛物型方程的Cauchy问题进行变量代换,我们把状态空间的维数降低了一维.为克服其中的奇异性问题,本文对方程进行了分解,第一部分的方程虽然保持奇性,但是其解具有一个精确表达式;而残差部分满足系数和初始条件都充分光滑的Cauchy问题,我们运用一般的差分方法对该部分进行了有效的数值计算。     

Schematic 4-designs

We study 4-designs with three intersection numbers. By the Cameron-Delsarte theorem, the blocks form a symmetric three-class association scheme. This imposes strong restrictions on the parameters of such designs. We calculate the eigenvalues of the association scheme from the design parameters and determine all admissible parameters with at most 1000 points. An infinite family of admissible parameters is discovered. Designs with small admissible parameters exist and are related to the quadratic residue codes.     

Residue minimization technique to analyze the efficiency of convective straight fins having temperature-dependent thermal conductivity

In this article, we propose and implement a numerical technique based on residue minimization to solve the nonlinear differential equation, which governs the temperature distribution in straight convective fins having temperature-dependent thermal conductivity. The form of temperature distribution is approximated by a polynomial series, which exactly satisfies the boundary conditions of the problem. The unknown coefficients of the assumed series are optimized using the Nelder–Mead simplex algorithm such that the squared L₂ norm of the residue attains its minimum value within a specified tolerance limit. The near-exact solution thus obtained is further used to calculate the fin efficiency. For the case of constant thermal conductivity, the obtained results are validated with the analytical solutions, while for the case of variable thermal conductivity, the obtained results are corroborated with those previously published in the literature. An excellent agreement in each case consolidates the effectiveness of the proposed numerical technique.     

Finite Laurent Developments and the Logarithmic Residue Theorem in the Real Non-analytic Case

This paper develops a general abstract non-holomorphic operator calculus under minimal regularity requirements on the family of operators through the concept of algebraic eigenvalue and the use of a, very recent, transversalization theory. Further, it analyzes under what conditions the inverse of a non-analytic family admits a finite Laurent development, and employs the new findings to calculate the multiplicity of a real non-analytic family through a logarithmic residue, so extending the applicability of the classical theory of I. C. Gohberg and coworkers. Applications to matrix families and Nonlinear Analysis are also explained.     

On the η‐function for bisingular pseudodifferential operators

In this work we consider the η‐invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace property for the Wodzicki residue of bisingular operators and show how the residues of the η‐function can be expressed in terms of the Wodzicki trace of a projection operator. Then we calculate the K‐theory of the algebra of 0‐order (global) bisingular operators. With these preparations we establish the regularity properties of the η‐function at the origin for global bisingular operators which are self‐adjoint, elliptic and of positive orders.     

Generation in singularity categories of hypersurfaces of countable representation type

The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category $$\textsf {D} _{\textsf {sg} }(R)$$ of a hypersurface R of countable representation type. For a thick subcategory $${\mathcal {T}}$$ of $$\textsf {D} _{\textsf {sg} }(R)$$ and a full subcategory $$\mathcal {X}$$ of $${\mathcal {T}}$$, we calculate the Rouquier dimension of $${\mathcal {T}}$$ with respect to $$\mathcal {X}$$. Furthermore, we prove that the level in $$\textsf {D} _{\textsf {sg} }(R)$$ of the residue field of R with respect to each nonzero object is at most one.     

Homotopy invariance of coherent Witt groups

 We prove the following general homotopy invariance theorem for coherent Witt groups: Let be a flat morphism of separated Gorenstein schemes of finite Krull dimension with affine fibers,i.e. π⁻¹(y) is an affine space over the residue field k(y) for all yY, then the induced homomorphism of coherent Witt groups is an isomorphism. As an application we calculate the (classical) Witt group of the affine hyperbolic sphere over a regular local ring. Received: 22 August 2001; in final form: 22 June 2002 / Published online: 1 April 2003     

广义Carmichael数

设n是一个合数,Z_n表示模n的剩余类环,r(x)∈Z_n[x]是一个首一的k(>0)次不可约多项式。本文引入n是k阶摸r(x)的Carmichael数的定义,全体这样的数记为集C_(k,r)(x),由此给出k阶Carmichael数集:C_k={∪C_(k,r)(x)|r(x)过全体Z_n上的首一k次不可约多项式}。显然C_1表示通常的Carmichael数集。作者得到了n∈C_(k,r(x))的一个充要条件,进而得到n∈C_k的一个充要条件及n∈C_2的一个更易计算的充要条件,还证明了C_1(?)C_2以及|C_2|=∞。